Newtonian mechanics describes motion using forces and acceleration, with the central law $\mathbf{F}=m\mathbf{a}$. It works extremely well for everyday motion, planetary orbits, and many engineering systems. However, its formulation becomes less elegant for constrained systems, generalized coordinates, and advanced fields like quantum mechanics, where action principles become more natural.

The Lagrangian is usually defined as $L = T - V$, where $T$ is kinetic energy and $V$ is potential energy. The actual path taken by a system makes the action $S = \int L\,dt$ stationary. From this principle, the Euler-Lagrange equations follow: $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right)-\frac{\partial L}{\partial q_i}=0.$$ This formulation handles constraints and generalized coordinates with unusual efficiency.

The Hamiltonian is typically the total energy expressed in terms of coordinates and conjugate momenta, $H(q_i,p_i,t)$. Hamilton's equations describe the evolution in phase space: $$\dot{q}_i = \frac{\partial H}{\partial p_i}, \qquad \dot{p}_i = -\frac{\partial H}{\partial q_i}.$$ This framework is especially useful for canonical transformations, statistical mechanics, and quantum theory.